Investigations Into the Effect of Spatial Correlation on Channel Estimation and Capacity of Multiple Input Multiple Output System

Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 doi:10.4236/ijcns.2009.24029 Published Online July 2009 (http://www.SciRP.org/journal/ijcns/).

Investigations into the Effect of Spatial Correlation on Channel Estimation and Capacity of Multiple Input Multiple Output System

Xia LIU1, Marek E. BIALKOWSKI2, Feng WANG1 1Student Member IEEE, School of ITEE, The University of Queensland, Brisbane, Australia 2Fellow IEEE, School of ITEE, The University of Queensland, Brisbane, Australia Email: {xialiu, meb, fwang}@itee.uq.edu.au Received December 17, 2008; revised March 28, 2009; accepted May 25, 2009

ABSTRACT

The paper reports on investigations into the effect of spatial correlation on channel estimation and capacity of a multiple input multiple output (MIMO) wireless communication system. Least square (LS), scaled least square (SLS) and minimum mean square error (MMSE) methods are considered for estimating channel properties of a MIMO system using training sequences. The undertaken mathematical analysis reveals that the accuracy of the scaled least square (SLS) and minimum mean square error (MMSE) channel estimation methods are determined by the sum of eigenvalues of the channel correlation matrix. It is shown that for a fixed transmitted power to noise ratio (TPNR) assumed in the training mode, a higher spatial correlation has a positive effect on the performance of SLS and MMSE estimation methods. The effect of accuracy of the estimated Channel State Information (CSI) on MIMO system capacity is illustrated by computer simulations for an uplink case in which only the mobile station (MS) transmitter is surrounded by scattering objects.

Keywords: MIMO, Channel Estimation, Channel Capacity, Spatial Correlation, Channel Modelling

1. Introduction been investigated. It has been shown that the accuracy of the investigated training-based estimation methods is In recent years, there has been a growing interest in mul- influenced by the transmitted power to noise ratio (TPNR) in the training mode, and a number of antenna tiple input multiple output (MIMO) techniques in rela- elements at the transmitter and receiver. In particular, it tion to wireless communication systems as they can sig- has been pointed out that when TPNR and a number of nificantly increase data throughput (capacity) without the antenna elements are fixed, the SLS and MMSE methods need for extra operational frequency bandwidth. In order offer better performance than the LS method. This is due to make use of the advantages of MIMO, precise channel to the fact that SLS and MMSE methods utilize the chan- state information (CSI) is required at the receiver. The nel correlation in the estimator cost function while the reason is that without CSI decoding of the received sig- LS estimator does not take the channel properties into nal is impossible [1–5]. In turn, an inaccurate CSI leads account. to an increased bit error rate (BER) that translates into a It is worthwhile to note that the channel properties are degraded capacity of the system [6–8]. governed by a signal propagation environment and spa- Obtaining accurate CSI can be accomplished using tial correlation (SC) that is dependent on an antenna con- suitable channel estimation methods. The methods based figuration and a distribution of scattering objects that are on the use of training sequences, known as the training- present in the path between the transmitter and receiver. based channel estimation methods, are the most popular. The works in [9,10] have demonstrated superiority of In [9,10], several training-based methods including least SLS and MMSE estimation methods, which make use of square (LS) method, scaled least square (SLS) method channel correlation, over the LS method neglecting and minimum mean square error (MMSE) method have channel properties. However, no specific relationship

Copyright © 2009 SciRes. Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 268 X. LIU ET AL. between spatial correlation and channel estimation accu- where HLOS denotes the LOS part as and HNLOS denotes racy has been shown. The works presented in [11,12] NLOS part. K is the Rician factor defined as the ratio of have reported on the relationship between spatial corre- power in LOS and the mean power in NLOS signal lation and estimation accuracy of MMSE method. How- component [17]. The elements of HLOS matrix can be ever, only simulation results, giving trends without any written as [18] further mathematical insight have been presented. rt 2 In this paper, we try to fill the existing void by pre- H LOS exp( Dj rt ) (3) senting the mathematical analysis explaining the effects  of channel properties on SLS and MMSE channel esti- where Drt is the distance between t-th transmit antenna and mation methods. It is shown that for a fixed TPNR, the r-th receive antenna. Assuming that the components of accuracy of SLS and MMSE methods is determined by NLOS are jointly Gaussian, HNLOS can be written as [19,20], the sum of eigenvalues of channel correlation matrix, 2/12/1  RHRH (4) which in turn characterizes the signal propagation condi- NLOS TgR tions. In addition, we report on the effect of spatial cor- where Hg is a matrix with i.i.d Gaussian entries. relation on both the channel estimation and capacity of Here, the Jakes fading model [21,22] is used to de- MIMO system. In the work presented in [13–16], it has scribe the spatial correlation matrices RR at the receiver been shown that the existence of spatial correlation leads and RT at the transmitter. An uplink case between a base to the reduced MIMO channel capacity. However, these station (BS) and a mobile station (MS) is assumed, as conclusions rely on the assumption of perfect CSI avail- shown in Figure 1. able to the receiver. In practical situations, obtaining The BS antennas are assumed to be located at a large perfect CSI can not be achieved. Therefore, in this paper height above the ground where the influence of scatterers we take imperfect knowledge of CSI into account while close to the receiver is negligible. In turn, MS is assumed evaluating MIMO capacity. to be surrounded by many scatterers distributed within a The rest of the paper is organized as follows. In Sec- “circle of influence”. For this case, the signal correlation BS tion 2, a MIMO system model is introduced. In Section 3, coefficients at the receiver BS and transmitter MS, ρR MS LS, SLS and MMSE channel estimation methods are and ρT , can be obtained from [22] and are given as: described and the channel estimation accuracy analysis is  MS() T  J [2/]T  (5) given. Section 4 shows derivations for the lower bound Tmn0 mn of MIMO channel capacity when the channel estimation 22  BS( R )Jj [R cos( )]exp(R sin( )) errors are included. Section 5 describes computer simu- Rmn 0mmnax mn lation results. Section 6 concludes the paper. (6) T R 2. System Description & Channel Model where, δmn and δmn are the antenna spacing distances between m-th and n-th antennas at transmitter and receiver, respectively; λ is the wavelength of the carrier; We consider a flat block-fading narrow-band MIMO sys- γmax is the maximum angular spread (AS); θ is the AoA tem with Mt antenna elements at the transmitter and Mr of LOS and J0 is the Bessel function of 0-th order. Using antenna elements at the receiver. The relationship between BS T MS R BS ρR (δmn ) and ρT (δmn ), the correlation matrices RR the received and transmitted signals is given by (1): MS and RT for BS and MS links can be generated as YHSV (1) s  BS() BS BS ( BS ) RR11  1Mr   where Ys is the Mr × N complex matrix representing the BS RR     (7) received signals; S is the Mt × N complex matrix repre-  BS BS BS BS  RM()1 RMM ( ) senting transmitted signals; H is the Mr × Mt complex  rr r channel matrix and V is the M × N complex zero-mean r white noise matrix. N is the length of transmitted signal. The channel matrix H describes the channel properties which depend on a signal propagation environment. Here, the signal propagation is modeled as a sum of the line of sight (LOS) and non-line of sight (NLOS) components. As a result, the channel matrix is represented by two terms and given as [17,18],

1 K HHNLOS HLOS (2) 11KK Figure 1. Jakes model for the considered MIMO channel.

Copyright © 2009 SciRes. Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 INVESTIGATIONS INTO THE EFFECT OF SPATIAL CORRELATION ON CHANNEL ESTIMATION 269 AND CAPACITY OF MULTIPLE INPUT MULTIPLE OUTPUT SYSTEM

MS MS MS MS 3.2. SLS Method TT()11   ()1M t MS The SLS method reduces the estimation error of the LS RT    (8) MS() MS MS ( MS )method. The improvement is given by the scaling factor TMtt1  TMMtγ which can be written as

tr{} RH 3. Training-Based Channel Estimation   (13) MSELS tr{} R H

For a training based channel estimation method, the rela- The estimated channel matrix is given as [9], [10] tionship between the received signals and the training ˆ tr{} RH † sequences is given by Equation (1) as H SLS  21H  YP (14)  nrMtrPP{( ) } trR { H }

YHPV (9) 2 Here, σn is the noise power; RH is the channel correla- H Here the transmitted signal S in (1) is replaced by P, tion matrix defined as RH=E{H H} and tr{.} implies the which represents the Mt × L complex training matrix trace operation. The SLS estimation MSE is given as [9,10] (sequence). L is the length of the training sequence. The 2 goal is to estimate the complex channel matrix H from ˆ MSESLS  E{} H H LS the knowledge of Y and P. F (15) 22 Here the transmitted signal S in (1) is replaced by P, (1 )tr { RH }  MSELS which represents the Mt × L complex training matrix (sequence). L is the length of the training sequence. The The minimized MSE of MMSE method can be written goal is to estimate the complex channel matrix H from as [9,10] the knowledge of Y and P. SLS MSELS tr{} R H The transmitted power in the training mode is assumed MSEmin  (16) MSELS tr{} R H to be constrained by P 2 where P is a constant and F  2 By taking into account expression (12), the minimized ||.||F stands for the Frobenius norm. According to [9,10], the estimation using LS, SLS or MMSE method requires MSE of the SLS method (16) can rewritten as orthogonality of the training matrix P. In the undertaken  MSE[( tr { R })11 ] analysis, the training matrix P is assumed to satisfy this SLS H MM2 condition. tr  [(tr { })1 ]1 (17) 3.1. LS Method 2 MMtr In the LS method, the estimated channel can be written n  [( )11 ] as [23],  i 2 i MMtr Hˆ  YP† (10) LS where n=min(Mr, Mt) and is λi the i-th eignvalue of the

† channel correlation RH. where {.} stands for the pseudo-inverse operation. If TPNR is fixed then the following equality can be The mean square error (MSE) of LS method is given derived as nn 11 2 MSE [( ) ]  (18) ˆ SLS iMM2  i MSELS  E{ H H LS } (11) iitr F As observed from (18), MSE decreases when the sum in which E{.} denotes a statistical expectation. According of eignvalues of R decreases. This shows that in order to [9,10], the minimum value of MSE for the LS method H to minimize MSE, the sum of eigenvalues of RH has to be is given as reduced. 2 LS M trM 3.3. MMSE Method MSEmin  (12)  In the MMSE method, the estimated channel matrix is in which ρ stands for transmitted power to noise ratio given as (19) [9,10,23], (TPNR) in training mode. Equation (12) indicates that Hˆ YP()HH RP 21 MI P R (19) the optimal performance of the LS estimator is not in- MMSE H n r H fluenced by channel matrix H. The MSE of MMSE estimation is given as

ˆ 2 4. MIMO Channel Capacity Taking into MSEMMSE  E{} H HMMSE  tr RE {} (20) F Account Channel Estimation Errors in which R is an estimation error correlation written as E The achievement of high channel capacity in a MIMO ˆˆH system depends on two factors. One is a rank of channel REHHE {(MMSE )( HH MMSE ) } (21) matrix or effectiveness of freedom (EDOF). The other ()RMPP121 H 1 Hnr one is the availability of CSI at the receiver. In [26,27,29] The minimized MSE is given as (22) [2,3,11] it has been shown that higher accuracy of CSI leads to higher channel capacity. However, the undertaken inves- 121HH  1 MSEMMSE tr{(  n M r Q PP Q)} (22) tigations have not considered the channel properties. If CSI is perfectly known at the receiver (but unknown In (22), Q is the unitary eigenvector matrix of RH and at the transmitter), the capacity of a MIMO system with Λ is the diagonal matrix with eigenvalues of R . The H Mr receive antennas and Mt transmit antennas can be minimized MSE for the MMSE method, given by Equa- expressed as [1,24,25], tion (22), can be rewritten using the orthogonality prop-  erties of the training sequence P and the unitary matrix Q, CE(log {det[ ISNR ( HHH )]}) (25) 2 M R as shown by M t

111 MSEMMSE  tr{(  Mr I)} In Equation (25), ρSNR is a signal to noise ratio (SNR). 111 The channel matrix H is assumed to be perfectly known ()01  M r  0 at the receiver. 0(111 M )  tr{2 r }In practical cases, H has to be replaced by the esti- 0 mated channel matrix, which carries an estimation error. 111 00( 2  M r )By assuming that the channel estimation error is defined n ˆ 111 as e and the estimated channel matrix as H ()irM i Hˆ  He (26) (23) The received signal can accordingly be written as, Assuming that TPNR in expression (13) is fixed, the bound for MSE is given by YHSeSV ˆ  (27) nn 111 Correlation of e is given as MSEMMSE ()i M r i (24) ii REHHHH{(ˆˆ )( )H }  2 I (28) The expression (24) shows that, similarly as in the E e SLS method, a smaller sum of eigenvalues of the channel 2 in which σe is the error variance. In [26,27], the defini- correlation RH leads to a smaller estimation error for the tion of error variance is slightly different. Using Equa- MMSE method. In other words, a smaller sum of eigen- tion (20), we have values of the channel correlation leads to the more accurate channel estimation. 2 MSE  e  (29) From the above mathematical analysis it becomes ap- M r parent that when the value of TPNR is fixed the accuracy of a training-based MIMO channel estimation is governed The channel capacity of MIMO system with an imper- by the sum of eigenvalues of the channel correlation matrix fectly known H at the receiver is defined as the maximum mutual information between Y and S and is given as RH. In turn, the properties of RH and its eigenvalues are determined by the channel properties which are influenced CISY max { ( ; ,Hˆ )} (30) by a signal propagation environment and an array antenna tr{} Q P elements and configuration. It is worthwhile to note that the spatial correlation (for If the transmitter does not have any knowledge of the example due to the presence of LOS component) is re- estimated channel, the mutual information in Equation sponsible for the channel rank reduction. In this case, the (30) can be written as [26–29], sum of eigenvalues of R has a smaller value. Thus from H I(;,SY Hˆˆˆ ) ISY (; | H ) hS ( | H ) hS ( | Y , Hˆ ) (31) the derived expressions, it is apparent that the spatial correlation (due to an increased LOS component) contributes Because adding any dependent term on Y does not in a positive manner to improving the training-based change the entropy [28], then MIMO channel estimation accuracy.

Copyright © 2009 SciRes. Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 INVESTIGATIONS INTO THE EFFECT OF SPATIAL CORRELATION ON CHANNEL ESTIMATION 271 AND CAPACITY OF MULTIPLE INPUT MULTIPLE OUTPUT SYSTEM

hSYH(|,)ˆ hS ( uYYH |,)ˆ (32) The lower bound of the ergodic channel capacity can be shown to be given as in which u is the MMSE estimator given as pHHˆˆH H CE{log [det( I )]} ESY{| Hˆ } 2 M r 22 u  (33) pen H ˆ EYY{| H} P s HHˆˆH M 1 Combining this with Equation (27), we have EI{log [det(t )]} 2 M r 22 nseP H 1 ESHS{(ˆˆ eSV ) | H } 2 u  M tn ˆˆH ˆ EHSeSVHSeSV{( )(  ) | H } ˆˆH SNR HH 1 ˆ H EI{log [det( )]} QH 2 M r M  MSE  t 1 SNR HQHˆˆHH E{} eQe 2 I nMr MMtr

(34) (40) Equation (40) indicates that for a fixed value of SNR, H where Q=E{SS } is a Mt by Mt correlation matrix of the capacity is a function of the estimated channel matrix 2 transmitted signal S defining the signal transmission Hˆ and the channel estimation error σe . As a result, the scheme. The autocorrelation matrix holds the property channel properties and the quality of channel estimation that trace(Q) equal to the total transmitted signal power influence the MIMO capacity. 2 Ps (ρSNR=Ps/σn ). If we assume the special case of Mt equal to Mr and the transmitted signal power being equally allocated to transmitting antennas, (34) becomes 5. Simulation Results

QHˆ H Here we present computer simulation results which u  ˆˆH 22 demonstrate the influence of channel properties on the HQH QeM I nM I rr (35) training-based channel estimation. A 4×4 MIMO system pHˆ H including 4-element linear array antennas both at the  transmitter and receiver is considered. The Jakes model pHHˆˆH  p22 I I eMrr nM presented in Section 2 is used to describe the propagation environment between BS and MS. The distance between in which p=Ps/Mt is the power allocated to the signal transmitter and receiver is assumed to be 100λ. The An- transmitted through each transmit antenna. Because con- gle of Arrival (AoA) of LOS is set to 0°. The training ditioning decreases the entropy therefore sequence length L is assumed to be 4. The default an- ˆˆ tenna element spacing at both BS and MS is set to 0.5λ hS(|)(|, uYH hS uYY H) (36) (wavelength). Figure 2 shows a relationship between MSE and the Then we have sum of eigenvalues of the channel correlation matrix for both MMSE and SLS methods. The results include an hS(|)(|, uYHˆ hS uYY Hˆ) (37) effect of maximum angle spread (AS), antenna spacing and Rician factor K, which are related to the sum of ei- In this case, genvalues of RH. The obtained results are given in four sub-figures A, B, C and D. I(;SY | Hˆˆ ) hS ( | H ) hS ( uYY | , Hˆ ) (38) Sub-figure A supports the theory presented in Section 3 that for MMSE and SLS methods channel estimation For the case of SH| ˆ and (|SuYYH ,ˆ ) having a errors are smaller for smaller sums of eigenvalues of RH. Gaussian distribution, (38) can be expressed as [26,27,29], When the sum of eignvalues increases, the channel estimation accuracy becomes worse. ˆ ISY( ; | H ) E {log2 [det( eQ )]} The relationship between the sum of eignvalues of RH H and the maximum AS is shown in sub-figure B. The sum EeESuYSuYH{log[det({()()|})  ˆ ]} 2 of eigenvalues increases when AS increases. Larger val- pHHˆˆH ues of AS correspond to a lower level correlation while EI{log [det( )]} 2 M r Ip()22 smaller values of AS correspond to a higher level corre- Menr lation. Higher values of spatial correlation lead to a (39) smaller sum of eigenvalues of RH. This condition helps to

Copyright © 2009 SciRes. Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 272 X. LIU ET AL. improve the accuracy of the training based MIMO chan- Figure 3 is plotted in three dimensions (3D) to provide nel estimation. a further support for the results of Figure 2. In this figure, Sub-figure C presents the relationship between the sum the relationship between MSE, TPNR ρ and K for of eigenvalues and the MS transmitter antenna spacing. MMSE method is presented at three different values of One can see that the sum of eigenvalues becomes smaller maximum AS (indicating three special correlation levels). when the spacing distance is less than 0.2λ. One can see that when TPNR is increased to 30dB the Sub-figure D gives the relationship between the sum estimation error decreases almost to zero. When the of eigenvalues and the Rician factor K. The sum of ei- value of Rician factor K is increased, indicating a genvalues is smaller at higher values of K (when the stronger LOS component in comparison with NLOS LOS component is strongest). This means that a stronger components, the MSE decreases. This is consistent with LOS component reduces the sum of eigenvalues and thus the trend observed in Figure 2 that a stronger LOS com- improves the channel estimation accuracy. ponent results in better estimation accuracy.

A. MSE vs Sum of channel correlation eigenvalues 1 B. Sum of eigenvalues vs Angle spread 20

0.5 SNR=5dB SNR=10dB 19 SNR=15dB MMSE MSE 0 16 16.5 17 17.5 18 18.5 19 18 Sum of eigenvalues 4.8 17 4.7 Sum of egnvaues of Sum

4.6 SNR=10dB 16

SLS MSE SLS 0 0.05 0.1 0.15 0.2 0.25 0.3 4.5 16 17 18 19 Ange Spread C. Sum of eigenvalues vs Antenna Spacing D. Sum of eigenvalues vs Racian factor K 20 19

18.5 19 18

18 17.5

RX spacing = 0.5λ 17 17 RX spacing = 1λ

Sum of eignvalues of Sum RX spacing = 2λ eignvalues of Sum 16.5

16 16 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 TX antenna spacing /λ K (dB)

Figure 2. Relationship between MSE vs Sum of eigenvalues of RH showing an im- pact of antenna spacing and the Rician factor K.

MMSE of 4x4 MIMO MSE vs ρ vs K vs AS

0.7

0.7 Angle Spread 0.2 rads 0.6

0.6 0.5 Angle Spread 0.1 rads 0.5

0.4 0.4 E Angle Spread 0.05 rads

MS 0.3 0.3

0.2 0.2

0.1 20 0.1 0 30 10 25 20 15 10 5 0 ρ (dB) K (dB)

Figure 3. 3D plot of MSE vs TPNR and K for different values of AS for MMSE method.

Copyright © 2009 SciRes. Int. J. Communications, Network and System Sciences, 2009, 4, 249-324 INVESTIGATIONS INTO THE EFFECT OF SPATIAL CORRELATION ON CHANNEL ESTIMATION 273 AND CAPACITY OF MULTIPLE INPUT MULTIPLE OUTPUT SYSTEM

The presented results also show that for MMSE figure B. method MSE is reduced for the smallest AS, which cor- From Figure 4, one can see that the mean square error responds to the highest level of spatial correlation. increases as the angle spread becomes larger. For all In the next step, we simulate the MIMO channel ca- three groups, at K factor of 20dB, MSE shows the best pacity under the condition of channel estimation error. performance while the worst accuracy occurs at the K Simulations are run for the cases of 2×2 MIMO and 4x4 factor of 0dB. In sub-figure B, the relationship trends are MIMO systems. The simulation settings including the different from the ones observed in sub-figure A. As the distance between the transmitter and receiver, training angle spread increases, the channel capacity is enhanced. sequence length, AoA, AS and antenna element spacing In all three groups of lines at three SNR values, the high- are same as in the earlier undertaken simulations. The est capacity occurs at K equal to 0dB while at 20dB the minimum mean square error (MMSE) channel estimation capacity is decreased. These two sub-figures indicate that method is applied for the Jakes model representing the the channel capacity is increased when the channel esti- channel between the BS and MS. The channel capacity is mation accuracy is reduced. This finding is opposite to determined using Equation (40). For simulation purposes, the one shown in [26,27,29]. This could be due to the RH is obtained using the actual channel matrix H and fact that in [26,27,29] the influence of spatial correlation TPNR is assumed to be equal to SNR. 100000 channel or K factor on the channel capacity was not considered. realizations are used to obtain the value of capacity. Fig- The presented simulation results strengthen the notion ure 4 shows the results for the 2×2 MIMO system. that the higher spatial correlation (due to a stronger LOS Simulation results are plotted in two-dimensions (2D) component) helps to improve the channel estimation ac- and include two sub-figures. Sub-figure A presents the curacy. At the same time it decreases the rank and EDOF relationship between MSE and angle spread (AS). There of the channel matrix. are three groups of lines drawn for three different values Similar findings are obtained for the 4×4 MIMO sys- of SNR. In each group, the lines correspond to three dif- tem, as illustrated in Figure 5. The results are shown for ferent values of K factor. The relationship between SNR of 5, 10 and 15dB and the Rician factor K of 0 and channel capacity and the angle spread is given in sub- 10dB.

A. 2x2A.2× MSE2 MSE v vss Angle Spread Spread

0.4 SNR=5dB K=0dB 0.3 K=0dB K=0dB K=10dB 0.2 K=10dB MSE

MSE SNR=10dB K=10dB K=20dB 0.1 K=20dB SNR=15dB K=20dB

0 0.05 0.1 0.15 0.2 0.25 0.3 AngleAngle SpSpr eadread

B. 2x2B. Channel2×2 Channel Cap Capacityacity vvss Angle Angle Spread Spread

7 SNR=15dB K=0dB ity 6 K=0dB K=0dB 5 SNR=10dB K=10dB K=10dB

4 K=10dB K=20dB

Channel Capacity SNR=5dB K=20dB Channel Capac 3 K=20dB

2 0.05 0.1 0.15 0.2 0.25 0.3 AngleAngle Spread

Figure 4. MSE vs Angle Spread and Channel estimation vs Angle Spread of a 2×2 MIMO under 3 different SNR and 3 different K factor.

A.A. 4x4 4× 4MSE MSE v vss AAngle ngle Spr Spread ead

0.8 SNR=5dB 0.6 K=0B K=0dB 0.4 K=0dB

MSE SNR=10dB K=10dB K=10dB 0.2 SNR=15dB K=10dB

0 0.05 0.1 0.15 0.2 0.25 0.3 AngleAngle Spread B. 4x4B. 4 ×Channel4 Channel Capacity Capacity vs AngleAngle Spread Spread

14 SNR=15dB 12 K=0dB K=0dB 10 K=0dB 8 SNR=10dB K=10dB

apacity K=10dB Capacity C 6 SNR=5dB K=10dB 4

2 0.05 0.1 0.15 0.2 0.25 0.3 AngleAngle SprSpread ead

Figure 5. MSE vs Angle Spread and Channel estimation vs Angle Spreadead of a 4×4 MMIMOIMO under 3 different SNR and 3 different K factor.

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